Now we consider two modules over $A$ and $B$, $M$ and $N$. We want to construct a map from $N$ to $M$. But the question is that the two modules are not over the same polynomial ring. So how can we make it?

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3

Please see the FAQ mathoverflow.net/faq for a list of sites where your question would be probably more appropriate.
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user2035Sep 17 '11 at 10:00

2

Have you tried to apply the "change of ring" technique ? For it, one uses a ring homomorphism $\varphi: R \to S$ and considers a $S$-Module $N$ as $R$-module in the obvious way. If $M$ is a $R$-module, then look for $R$-module homomorphisms $M \to N$. In your example $\varphi$ can be taken to be the inclusion $A \hookrightarrow B$.
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Todd Leason Sep 17 '11 at 12:29

I doubt that there is any site where this question would be appropriate.
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Steven LandsburgSep 17 '11 at 13:43